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Originally posted by LemonJelloAfter thinking for a while I think you've at least partially changed my mind. It helps to have an example when talking about this stuff, so I thought of a few things I can be said to know infallibly. The two examples of my infallible knowledge I'll use are "I exist" and the law of non-contradiction.I think we agree that, since knowledge is factive, to say that I know something already implies the something is true. So the adjective infallible doesn't add anything in a sentence fragment like "God's infallible knowledge ...". So I feel that it must apply to an agent making assertions or as an adverb to the making of the assertions.
...[text shortened]... be wrong? If such further constraints are intended, then modality is an integral part of this.
I know I exist because I believe I do, its true that I exist by any sane truth test, and the justification is that I have experience of qualia and so forth. This isn't to say I couldn't be a brain in a vat or a simulation or some such, in that case I'd still exist but the nature of my existence wouldn't be what I thought it was. In the case of the law of non-contradiction the justification is a matter of logical proof and therefore rock solid.
This led me to ask what I'm applying the predicate of infallibility to. It cannot be applied to me globally, because I'm perfectly capable of making incorrect statements - heck look at what's been happening to my rating recently! If infallibility applies to the justification of the belief and we have knowledge as justified true belief, then knowledge can be described as infallible. My only quibble is that sentences like "I am infallible in this matter." make sense so I can still claim that it is the agent and not the knowledge itself that is infallible even if the agent is not always infallible.
When I was constructing my definition of infallibility I wanted it as free as possible. I didn't want to exclude lucky guessers. I was thinking that it would be straightforward to add a modal operator to capture that case. Although adding a modal operator isn't necessarily the right thing to do. Consider the following:
Suppose Alice is one of the Pythia. In one possible world Alice is infallible because Apollo answers the questions put to her. Apollo only exists in this one possible world. In another possible world she's just repeatedly lucky. In all other possible worlds she gets it wrong sometimes. For the two infallible Alices my formula captures their infallibility and excludes fallible Alice. We want it to separate the two infallible ones into the weakly and strongly infallible Alices.
∀x (S(x)&B(x,a) -> □T(x)) <-> ¬F(a)
seems right as we have the necessity of the truth condition, but only works if Alice is infallible in all possible worlds. But I have her infallibility contingent on Apollo existing. So the problem is that I'm trying to add a modal operator to rules in individual possible worlds which breaks possible world semantics, at least to my rather limited understanding of it. To capture the case where Alice is only infallible in one possible world I've got to drop the modal operator and have:
∀x A&(S(x)&B(x,a) -> T(x)) <-> ¬F(a)
Where A means Apollo exists and talks to Alice. Which means that when infallibility is contingent on something I can't use a modal operator (?) and have to include some sort of proposition to ensure the existence of the thing infallibility is contingent on. When it is not contingent then I can use a modal operator.
This leads me to think that the condition for infallible knowledge should be something like:
K(P) -> □P
or
□(K(P) -> □P)
P is some proposition and K(P) means some agent infallibly knows P. But I'm too much of a beginner with modal logic to cope with the different axiomatic systems and when they're applicable. For example whether I can have necessity within a single possible world.
Originally posted by twhiteheadThe coin may have come up tails. It was possible for the coin to come up tails and so we imagine an alternative universe, like this one in all other respects where it didn't.
Which is why I earlier stated that the issue may not be one of logic at all.
[b]Suppose we toss a coin and it comes up heads. What do we mean when we say something like: "The coin was not necessarily going to come up heads."?
I don't know. In the case of a single universe with a single timeline, you are not making sense. The coin did come up head ...[text shortened]... happen. So making precise statements about them is a useless exercise (as well as being wrong).[/b]
The term they use is actual world. I'm certain the real world exists. The actual world is the one we are in and definitely exists. The others are called possible worlds because it is possible that the actual world could be identical in all respects to one of them, if it is identical in all respects it is that world.
What is a timeline? As far as I know this is a term from Star Trek, I've never heard it in any other context than Star Trek. It's certainly not used in physics, and I've not noticed philosophers use it.
Possible worlds are things we imagine. We can imagine as many worlds as we like. This is a well established method in philosophy, so I'd be wary about making statements like: "It's useless and wrong.". My description of it probably wasn't very good, so try reading the Wikipedia page.
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Originally posted by DeepThoughtSo can I, or can I not, claim to have known that it would come up heads? I just want to understand what you mean by 'knowledge'.
The coin may have come up tails. It was possible for the coin to come up tails and so we imagine an alternative universe, like this one in all other respects where it didn't.
The term they use is actual world. I'm certain the real world exists. The actual world is the one we are in and definitely exists.
But does the future exist? If tomorrow I throw a coin and it comes up heads, can we today say that in the real world, I will get heads tomorrow and all other worlds though 'possible' are not 'actual'? Or can we only say that tomorrow after the event?
The others are called possible worlds because it is possible that the actual world could be identical in all respects to one of them, if it is identical in all respects it is that world.
I can't seem to wrap my head around that. I think it needs more expansion to make sense.
What is a timeline?
One suggestion in physics is that there are two futures, both of which are real, in one I throw a heads, in another, I throw a tails. In each there will be an instance of me going 'Hey, I'm in the one where I threw a ....'.
The other possibility is that there is only one real future in which I throw say heads, and no future in which I throw tails.
Originally posted by twhiteheadFor knowledge, that's to say knowledge of, we mean justified true belief. The justification is required to avoid several traps to do with being right for the wrong reasons [1]. There is also knowledge how (I know how to bake a cake) and acquaintance knowledge (I know Fred), the justified true belief formulation does not apply to them [2].
So can I, or can I not, claim to have known that it would come up heads? I just want to understand what you mean by 'knowledge'.
[b]The term they use is actual world. I'm certain the real world exists. The actual world is the one we are in and definitely exists.
But does the future exist? If tomorrow I throw a coin and it comes up heads, can we ...[text shortened]... there is only one real future in which I throw say heads, and no future in which I throw tails.[/b]
There are two positions one called eternalism in which the past and future exist, the other presentism in which only the present exists [2]. I am an eternalist with respect to the past. With the future I'm not sure, but General Relativity (which is just a theory) it seems to me depends on this. If the future is not set then you can argue that it doesn't exist, but you can also argue that it does exist its just not set yet. It depends on what we mean when we say it exists.
This is what identity means [3]. If two objects have all the same properties, including position then they are the same object. Consider 2 + 2 = 4. The result of the arithmetic operation of addition is a number which is the successor of the successor of the successor of one, which is how four is defined in Presburger arithmetic [4]. We do not have multiple versions of 4 to cope with the different ways of reaching it, there is only one 4.
There is an objection to this from quantum mechanics. Two electrons are forbidden to be in the same state so the problem doesn't arise, one can always distinguish electrons by their quantum numbers, this is the content of Pauli's exclusion principle and chemistry would be very different without it. With photons they are bosons and are allowed to be in the same state. However the paradigm theory at the moment is quantum field theory and in that a photon is a quantum of a field. So if two quanta are in the same state then it just means the excitation is bigger, there aren't really two photons, there is an electro-magnetic field with two field quanta. The field quanta are a way of describing magnitude and shouldn't be considered objects.
I spent a considerable chunk of my life studying physics and the terms was never used. However it is possible to do that and miss a term in use. Also it's possible it was introduced into physics by a trekkie. What you have described sounds like possible worlds to me with an additional constraint that the possible worlds are required to be identical to the actual world up to a particular time and can differ after that. In Everett's many worlds interpretation of quantum mechanics something very like possible worlds exist. However in that the laws of physics are required to be the same in all of the worlds. In the philosophical possible worlds they can change, or more likely just ignore, the laws of physics.
[1] http://en.wikipedia.org/wiki/Gettier_problem
[2] http://en.wikipedia.org/wiki/Epistemology
[3] http://en.wikipedia.org/wiki/Philosophy_of_space_and_time
[4] http://en.wikipedia.org/wiki/Identity_(philosophy)
[5] http://en.wikipedia.org/wiki/Presburger_arithmetic
The reason I chose Presburger arithmetic is that it is very simple and immune to Gödel's incompleteness theorem so the statement is completely provable in first order logic. We know this one infallibly. ZFC based definitions suffer from the problem that ZFC cannot be both consistent and complete, it's complicated enough for Gödel's incompleteness theorem to apply and so my argument would depend on a system that is not internally provably consistent.
Here's the two Wikipedia pages on modal logic and possible world semantics. The one on modal logic is not an easy read.
[6] http://en.wikipedia.org/wiki/Modal_logic
[7] http://en.wikipedia.org/wiki/Possible_world
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Originally posted by DeepThoughtFAO LemonJello
After thinking for a while I think you've at least partially changed my mind. It helps to have an example when talking about this stuff, so I thought of a few things I can be said to know infallibly. The two examples of my infallible knowledge I'll use are "I exist" and the law of non-contradiction.
I know I exist because I believe I do, its true th ...[text shortened]... en they're applicable. For example whether I can have necessity within a single possible world.
On further reflection, and having read the page on coherentism in the meantime, then if the coherentists are right your formula for infallibility works. Since then the condition for omnipotence implies a complete and therefore perfectly coherent all embracing body of knowledge and so implies infallibility, but I feel doesn't adequately describes it. It's a sufficient condition rather than either a necessary condition or a particularly descriptive one. We still don't need a modal operator though. A universal quantifier helps to reinforce the statement:
∀P (P -> K(P))
would signify omnipotence and therefore infallibility without the use of the necessity operator.
I'm worried about the modal operators in my earlier attempt at strong infallibility. The problem is that I want to be able to use necessary within a single possible world. I can do that by making a set of sets of possible worlds and have the necessity operator operating in one of the subsets, but this seems clumsy.
I want to be able to assert Alice's infallibility without specifying the reason for her infallibility and without making her necessarily infallible in order to cover the case where she is contingently, but strongly, infallible.
Originally posted by twhiteheadA contingent truth is one that, although true, could possibly have been false. A necessary truth, on the other hand, is one that could not possibly have been false. So, do you believe that we can have knowledge of contingent truths?
Please define 'contingent truth'. Are you saying for example: I claim to know that my next coin toss will be heads, then after I toss the coin and it turns out to be heads I announce 'I knew that!'?
Originally posted by bbarrI believe that we typically use the word 'knowledge' to include contingent truths. Whether we can or can not 'know' contingent truths is not a matter of belief but of definition.
A contingent truth is one that, although true, could possibly have been false. A necessary truth, on the other hand, is one that could not possibly have been false. So, do you believe that we can have knowledge of contingent truths?
But I would also like to question the whole idea of the concept of 'possibly'. What does it mean? If the future exists, then there are no contingent truths to be known. All truths are necessary truths.
This whole discussion is mired by our intuitive belief in an unfixed future and it is very difficult to throw off that belief even speculatively.
Take a book such as the Lord of the Rings. We know that Frodo goes to Mordor. Could it have been different? The book is written. It is fixed. Frodo goes to Mordor. Is that a contingent truth or not?
Originally posted by DeepThought
After thinking for a while I think you've at least partially changed my mind. It helps to have an example when talking about this stuff, so I thought of a few things I can be said to know infallibly. The two examples of my infallible knowledge I'll use are "I exist" and the law of non-contradiction.
I know I exist because I believe I do, its true th ...[text shortened]... en they're applicable. For example whether I can have necessity within a single possible world.
This leads me to think that the condition for infallible knowledge should be something like:
K(P) -> □P
P is some proposition and K(P) means some agent infallibly knows P
This condition, when conjoined with the idea that some agent such as God has infallible foreknowledge of our actions, leads directly to fatalism.
To see this, let P be a proposition regarding future action (such as that S will do A). Then:
(1)K(P) -> Necessarily P. [Your condition above]
(2)K(P). [Supposition of infallible foreknowledge]
(3)Necessarily P. [From (1) & (2) ]
(4)It is not possible that ~P. [Just restatement of (3) ]
(5)Hence S is not free with respect to A. By extension, libertarianism is false.
Also, in a previous thread, bbarr presented a very perspicuous version of the argument that shows that fatalism follows readily from this sort of infallibility condition. So, this infallibility condition would be a major problem for the libertarian theist who claims that God has infallible foreknowledge of our actions. However, it seems doubtful that this theist would need to be committed to such a condtion.
It seems several in this thread are also of the intuition that the necessity of P should follow from the fact that P is infallibly known. But, intuition aside, what actual argument shows this? And what is the type of necessity at issue? If it is a matter of logical necessity, then there should be a contradiction entailed by the conjunction of the following:
(a)K(P).
(b)Possibly ~P.
Whether or not a contradiction follows from (a) & (b) depends on how we unpack (a) . According to the condition you gave above, a contradiction follows readily from (a) & (b) . However, I am not at all convinced this is a fair condition to impose on the libertarian theist. It seems to me that, upon unpacking infallibility regarding God's cognition, all this theist really needs is something like the following (and note here that modality will be integral to this, to ensure that we are intending a strong sense of infallibility):
(a1) God believes P.
(a2) It is not possible both that God believes P and that ~P.
(b)Possibly ~P. [Or in other words, ~(Necessarily P)]
Is there a logical contradiction here? No there is not. So it seems clear to me that arguments for logical fatalism will fail. However, I would add two further points here.
First, there could still be some sort of necessity other than logical necessity – let us call it *Necessity* -- such that from (a1) & (a2) it follows that P is *Necessary* in a way that still itself precludes S from being free with respect to A. Those arguments are worth exploring.
Second, one needs justificatory conditions to bridge the gap from belief to foreknowledge, and it could be that there are no conceivable justificatory conditions such that one could foreknow that P while having no possibility of epistemic error. That would be worth exploring too, I think.
Originally posted by DeepThought
FAO LemonJello
On further reflection, and having read the page on coherentism in the meantime, then if the coherentists are right your formula for infallibility works. Since then the condition for omnipotence implies a complete and therefore perfectly coherent all embracing body of knowledge and so implies infallibility, but I feel doesn't adequately ...[text shortened]... arily infallible in order to cover the case where she is contingently, but strongly, infallible.
We still don't need a modal operator though. A universal quantifier helps to reinforce the statement:
∀P (P -> K(P))
would signify omnipotence and therefore infallibility without the use of the necessity operator.
You stated that K(P) stands for the proposition that some agent infallibly knows P. Your encoding here only gets you that all true propositions are infallibly known. But that just begs the question of in what exactly this infallibly knowing consists. Again, I am skeptical you can give any formulation of a strong sense of infallibility without, at the end of the day, modality. After all, you need to insure against the mere possibility of epistemic error.
It seems to me that something like "Necessarily, if P then G knows P" gets the job done, and in compact form. It implies that G is omniscient, which is something you already conceded on page 4. But it also gets that G is infallible and in a strong sense, since it is covering all possible worlds. For example, there will be no possible world wherein both G believes P and ~P.
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Originally posted by LemonJelloYes, I managed to impose fatalism despite having read bbarr's argument and having had him go through essentially the same point regarding predestination. Insufficient self-criticism of my train of thought while writing the post I'm afraid. Although in my defence it's an internet forum and the medium tends to cause that.
This leads me to think that the condition for infallible knowledge should be something like:
K(P) -> □P
P is some proposition and K(P) means some agent infallibly knows P
This condition, when conjoined with the idea that some agent such as God has infallible foreknowledge of our actions, leads directly to fatalism.
To see ...[text shortened]... t P while having no possibility of epistemic error. That would be worth exploring too, I think.
For *necessity* are you thinking of something nomological? For an alternative word to avoid the name space collision how about required? It's not quite got the same meaning, but it means we can avoid using the same word.
One of my opponents, Peter Seery, was wondering about the quote from the Hitchhiker's Guide on my profile and in the ensuing discussion he mentioned a nice example of foreknowledge which doesn't imply causation, which he read in a book on Bahai. Namely that we know that the sun will come up tomorrow. The level of justification for that is pretty immense - assuming NASA et al aren't keeping something from us - to the extent that it's pretty close to infallible.
Edit:
Regarding your second post, I think the pennies finally dropped. Using this sun coming up example, it's not infallible because there's a possible world where a large, fast moving lump of dark matter is going to smash the earth to smithereens [1] and the sun won't come up (even on a dead planet) because the world will be broken to bits - there won't be anything for the sun to come up on. So the necessity operator in □(K(P) <-> P) assures us that the omniscient entity must have infallible justification as the knowledge is protected against her ever being wrong.
Thanks for your patience.
[1] I can't think of any physically realistic scenario for this where we wouldn't have forewarning - if it only interacts gravitationally then it might kill everyone but it's difficult to see it causing the earth to stop spinning - and if it's a black hole we'd see X-rays and so forth. It's borderline impossible. I'm not sure if this possible world exists, but it was helpful for the point.
Originally posted by DeepThoughtBut the sun coming up is caused. And we can even come very close to claiming that it is caused by your knowledge.
One of my opponents, Peter Seery, was wondering about the quote from the Hitchhiker's Guide on my profile and in the ensuing discussion he mentioned a nice example of foreknowledge which doesn't imply causation, which he read in a book on Bahai. Namely that we know that the sun will come up tomorrow. The level of justification for that is pretty immens ...[text shortened]... ASA et al aren't keeping something from us - to the extent that it's pretty close to infallible.
A universe in which you can have that level of certainty about the sun coming up is a universe in which the sun will come up.
In addition, a universe in which you have have that level of certainty about the sun coming up is a universe in which things are at least partially deterministic. The sun has no free will.
Originally posted by twhiteheadThe sun coming up is caused by it continuing to shine and conservation of the Earth's angular momentum and nothing coming along to stop that. My near infallible knowledge that it is going to come up is not what causes it to. Understanding the causal relationship is what gives me the justification for my belief that it's going to rise, which causes the true belief to be knowledge. But the justification is in the causal relationship, it is not the causal relationship.
But the sun coming up [b]is caused. And we can even come very close to claiming that it is caused by your knowledge.
A universe in which you can have that level of certainty about the sun coming up is a universe in which the sun will come up.
In addition, a universe in which you have have that level of certainty about the sun coming up is a universe in which things are at least partially deterministic. The sun has no free will.[/b]
Why bring free will into it? My point was just to do with knowledge. If we want to be able to discuss things that can be known in advance infallibly, or near infallibly, then it will help to have some examples. This is a fairly useful example. Assuming you read LJ's post then having some criteria for infallibility help discuss whether it is logically possible to know something infallibly before it's happened. As things stand we have infallibility described by the necessity operator, which does the job, but doesn't tell us anything about the justificatory process, except that it's infallible.
The sun coming up is quite helpful, because it relies on some laws of physics and no sneak attacks by dark matter. So we have criteria for the justification to be flawed. This suggests to me that coherentism should be a criterion for infallibility, since we have to control for confounders and knowing that they are absent is part of the justification for the knowledge that the sun is going to come up.
However, there is a danger that one ends up having to know everything. To be certain about the confounders one has to be certain there are no confounders for the confounders and so on until we need to know everything that's happening within a 50,000 light seconds of the earth and the internal state of the sun (just in case our theories of stars are wrong and it can supernova after all).
Originally posted by LemonJelloSomething's occurred to me about discussions concerning free will using possible worlds. Suppose I have some decision to make, call it D. In the actual world I go for ¬D. There's some possible world where I do D. Now, if I knew about the other me that did that I might say: "I wouldn't do that, that's not like me". So I'm wondering to what extent my counterpart (to borrow the language of modal realism) in the other world can be called me?
This leads me to think that the condition for infallible knowledge should be something like:
K(P) -> □P
P is some proposition and K(P) means some agent infallibly knows P
This condition, when conjoined with the idea that some agent such as God has infallible foreknowledge of our actions, leads directly to fatalism.
To see ...[text shortened]... t P while having no possibility of epistemic error. That would be worth exploring too, I think.
In a more extreme case suppose the decision is whether to go swimming or not. We can imagine a possible world where as a child I'd had some trauma where I'd nearly drowned. In that possible world I am so scared of swimming that I can't meaningfully be described as having free will in the matter. I don't want to swim in that world, so I don't even have a decision to make. But in that possible world my counterpart is traumatised and has a personality difference and so isn't me.
So suppose a possible world that is identical to this world in all respects, up until the moment I'm faced with the decision. If it's identical until that moment then it seems reasonable to think I'll make the same decision as in this world, but then by the law of identity I'm describing the actual world. So I haven't actually looked at the case of a different possible world.
Which leaves us trapped between two potential fallacies, equivocation on the one hand and begging the question on the other.
Originally posted by DeepThoughtAnd I am challenging your understanding of causal relationships. If there are many possible universes, then the ones in which you know the sun will come up are ones such that the sun will come up. Why can we not say that your knowing causes the sun to come up?
The sun coming up is caused by it continuing to shine and conservation of the Earth's angular momentum and nothing coming along to stop that. My near infallible knowledge that it is going to come up is not what causes it to. Understanding the causal relationship is what gives me the justification for my belief that it's going to rise, which causes the ...[text shortened]... he justification is in the causal relationship, it is not the causal relationship.
Why bring free will into it?
Because the whole discussion relates to free will. If your knowing that the sun will come up demonstrates that the sun has no free will, then your knowing what I will do tomorrow, may demonstrate that I too have no free will.
In fact your knowing that the sun will come up is at least partly predicated on your belief that I will not blow up the sun tonight.
Originally posted by twhiteheadOk., it's not April the 1st, so, is this an actual position you're taking?
And I am challenging your understanding of causal relationships. If there are many possible universes, then the ones in which you know the sun will come up are ones such that the sun will come up. Why can we not say that your knowing causes the sun to come up?
[b]Why bring free will into it?
Because the whole discussion relates to free will. If yo ...[text shortened]... ll come up is at least partly predicated on your belief that I will not blow up the sun tonight.[/b]